Question

A \(10 \mathrm{MHz}\) uniform plane wave having an initial average power density of \(5 \mathrm{~W} / \mathrm{m}^{2}\) is normally incident from free space onto the surface of a lossy material in which \(\epsilon_{2}^{\prime \prime} / \epsilon_{2}^{\prime}=0.05, \epsilon_{r 2}^{\prime}=5\), and \(\mu_{2}=\mu_{0} .\) Calculate the distance into the lossy medium at which the transmitted wave power density is down by \(10 \mathrm{~dB}\) from the initial \(5 \mathrm{~W} / \mathrm{m}^{2}\).

Short answer

Question: Calculate the distance into the lossy medium at which the transmitted wave power density is down by 10 dB from the initial power density. Answer: To find the distance into the lossy medium, follow these steps: 1. Calculate the reflection coefficient (R) at the interface between the two media. 2. Calculate the transmitted power density (T) at the surface of the lossy material. 3. Determine the attenuation constant (α) in the lossy medium. 4. Calculate the distance (x) into the lossy medium for the required 10 dB power density reduction. Using the given information and equations, compute the distance x by substituting all the known values, including T and α, into the final equation for x.

Step-by-step solution

Calculate the Reflection Coefficient (R)

First, we have to find the reflection coefficient at the interface between the two media. This can be calculated using the formula: \(R = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}\) where \(\eta_1\) is the intrinsic impedance of free space (approximately 120π Ω), \(\eta_2\) is the intrinsic impedance of the lossy material, and R is the reflection coefficient. Since the magnetic permeability of the lossy medium is equal to that of free space (\(\mu_2 = \mu_0\)), the relative permeability \(\mu_r = \frac{\mu_2}{\mu_0} = 1\). So, we can write: \(\eta_2 = \sqrt{\frac{\mu_2}{\epsilon_2'}} = \sqrt{\frac{\mu_0}{\epsilon_0 \epsilon_{r2}'}} = \frac{\eta_1}{\sqrt{\epsilon_{r2}'}} \) Now, we can find the reflection coefficient: \(R = \frac{\frac{\eta_1}{\sqrt{\epsilon_{r2}'}} - \eta_1}{\frac{\eta_1}{\sqrt{\epsilon_{r2}'}} + \eta_1}\)

Calculate the Transmitted Power Density (T)

Next, we need to calculate the transmitted power density at the surface of the lossy material. This can be found using the formula: \(T = S_1(1 - |R|^2)\) where \(S_1\) is the initial average power density (\(5 \mathrm{~W}/\mathrm{m}^2\)) and T is the transmitted power density.

Determine the Attenuation Constant (α)

We have to find the attenuation constant in the lossy medium. This can be calculated using the formula: \(\alpha = \omega \sqrt{\frac{\mu_2 \epsilon_2''}{2}} = 2\pi f \sqrt{\frac{\mu_0 \epsilon_0 \epsilon_{r2}''}{2}}\) We know \(\epsilon_{r2}'' = { 0.05 \times \epsilon_{r2}' }\)

Calculate the Distance in the Lossy Medium

Now we are ready to calculate the distance into the lossy medium at which the transmitted wave power density is down by 10 dB from the initial power density. The power reduction in dB can be expressed as: \(10 \mathrm{~dB} = 10 \log{\frac{T}{S_2}}\), where \(S_2\) is the transmitted power density at the required distance. Now we can rewrite the equation in terms of the distance in the lossy medium: \(x = \frac{1}{\alpha} \left[{10 \log{({\frac{T}{S_2}}) \frac{1}{10}} }\right]\) Now substitute the values of T and α in the equation to calculate the distance x.

Key concepts

Reflection Coefficient

When a plane wave encounters a boundary between two different materials, part of the wave is reflected, and part is transmitted. To understand how much of the wave will be reflected, we use the reflection coefficient, denoted as \( R \). This coefficient is a measure of how much power is reflected relative to the incident power. It's determined by the intrinsic impedance of the two media involved.

• The intrinsic impedance of free space (\( \eta_1 \)) is about \(120\pi\) ohms.
• The impedance of the lossy material (\( \eta_2 \)) considers the dielectric properties and is given by \( \eta_2 = \sqrt{\frac{\mu_0}{\epsilon_0 \epsilon_{r2}'}} \).

The reflection coefficient is then calculated using the formula:\[ R = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1} \]This equation shows that the reflection coefficient depends significantly on the differences in impedance between the two media, which is influenced by factors like the relative permittivity.

Attenuation Constant

In a lossy medium, energy from an electromagnetic wave is absorbed, causing the wave to weaken as it propagates. The rate at which this power reduction happens with distance is characterized by the attenuation constant, represented by \( \alpha \). The attenuation constant is given by:\[ \alpha = \omega \sqrt{\frac{\mu_2 \epsilon_2''}{2}} \]where:

• \( \omega \) is the angular frequency of the wave, \( \omega = 2\pi f \)
• \( \epsilon_2'' \) is the imaginary part of the permittivity, indicating material losses

It's essential to note that as \( \epsilon_2'' \) increases, so does \( \alpha \), signifying more rapid attenuation. Lossy materials have both a real part (\( \epsilon' \)) affecting capacitance and an imaginary part (\( \epsilon'' \)) depicting energy losses. Understanding these components helps in determining how quickly the power of an electromagnetic wave is absorbed by a lossy medium.

Power Density

Power density describes how much power passes through a unit area in the direction of energy flow of an electromagnetic wave. It is measured in watts per square meter (\( W/m^2 \)). Initially, the wave has a known power density as it hits the interface from free space.When considering the transmitted wave, the power density changes as some energy reflects back. The transmitted power density \( T \) can be found using:\[ T = S_1(1 - |R|^2) \]where:

• \( S_1 \) is the initial power density.
• |R| represents the magnitude of the reflection coefficient.

The equation implies that only a portion of the initial power density is transmitted, depending on the reflection coefficient. As a wave moves deeper into the lossy material, the power density diminishes further due to attenuation.

Lossy Material Properties

Lossy materials are characterized by their ability to absorb electromagnetic energy and convert it into other forms, like heat. This absorption results in energy loss of the propagating wave. Key properties that describe lossy materials include:

• Loss Tangent: The ratio \( \tan(\delta) = \frac{\epsilon''}{\epsilon'} \) is a measure of how "lossy" a material is.
• Complex Permittivity: This is expressed as \( \epsilon = \epsilon' - j \epsilon'' \), where \( \epsilon' \) is the real part dictating electric field storage capability, while \( \epsilon'' \) governs energy dissipation.
• Magnetic Permeability: In this context, it’s given as \( \mu_{2} = \mu_{0} \), meaning the material's permeability matches that of free space, simplifying calculations.

These properties critically affect how electromagnetic waves behave as they travel through the material, dictating aspects like the attenuation rate and depth to which waves can penetrate before their power density drops significantly.